Travelling waves in a complex Ginzburg-Landau equation with time-delay feedback

One of the simplest ways to create nonlinear oscillations is the Hopf bifurcation. The spatiotemporal dynamics observed in an extended medium with diffusion (e.g., a chemical reaction) undergoing this bifurcation is governed by the complex Ginzburg-Landau equation, one of the best-studied generic models for pattern formation, where besides uniform oscillations, spiral waves, coherent structures and turbulence are found. The presence of time delay terms in this equation changes the pattern formation scenario, and different kind of travelling waves have been reported. In particular, we study the complex Ginzburg-Landau equation that contains local and global time-delay feedback terms. We focus our attention on plane wave solutions in this model. The first novel result is the derivation of the plane wave solution in the presence of time-delay feedback with global and local contributions. The second and more important result of this study consists of a linear stability analysis of plane waves in that model. Evaluation of the eigenvalue equation does not show stabilisation of plane waves for the parameters studied. We discuss these results and compare to results of other models.

Surface wave techniques for the study of engineering structures and materials

A wire drive pulse echo method of measuring the spectrum of solid bodies described. Using an 's' plane representation, a general analysis of the transient response of such solids has been carried out. This was used for the study of the stepped amplitude transient of high order modes of disks and for the case where there are two adjacent resonant frequencies. The techniques developed have been applied to the measurenent of the elasticities of refractory materials at high temperatures. In the experimental study of the high order in-plane resonances of thin disks it was found that the energy travelled at the edge of the disk and this initiated the work on one dimensional Rayleigh waves.Their properties were established for the straight edge condition by following an analysis similar to that of the two dimensional case. Experiments were then carried out on the velocity dispersion of various circuits including the disk and a hole in a large plate - the negative curvature condition.Theoretical analysis established the phase and group velocities for these cases and experimental tests on aluminium and glass gave good agreement with theory. At high frequencies all velocities approach that of the one dimensional Rayleigh waves. When applied to crack detection it was observed that a signal burst travelling round a disk showed an anomalous amplitude effect. In certain cases the signal which travelled the greater distance had the greater amplitude.An experiment was designed to investigate the phenanenon and it was established that the energy travelled in two nodes with different velocities.It was found by analysis that as well as the Rayleigh surface wave on the edge, a seoond node travelling at about the shear velocity was excited and the calculated results gave reasonable agreement with the experiments.